Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

ZHOU Jundong

doi:10.3969/j.issn.1000-5641.201911009

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ZHOU Jundong. Rigidity of submanifolds with parallel mean curvature in a hyperbolic space[J]. Journal of East China Normal University (Natural Sciences), 2020, (2): 8-14. doi: 10.3969/j.issn.1000-5641.201911009

Citation:

ZHOU Jundong. Rigidity of submanifolds with parallel mean curvature in a hyperbolic space[J]. Journal of East China Normal University (Natural Sciences), 2020, (2): 8-14. doi: 10.3969/j.issn.1000-5641.201911009

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Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

doi: 10.3969/j.issn.1000-5641.201911009

ZHOU Jundong

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei　230026, China
2.
School of Mathematics and Statistics, Fuyang Normal University, Fuyang Anhui　236037, China

Received Date: 2019-02-22
Publish Date: 2020-03-01

Abstract

Abstract

Let $ M $ be a complete submanifold with parallel mean curvature in a hyperbolic space and $ \Phi $ be the traceless second fundamental form of $ M $. In this paper, it is shown that the submanifold is totally umbilical if the $ L^2 $ norm of $ |\Phi| $ has less than quadratic growth on any geodesic ball of $ M $ and either $ \sup_{x\in M}|\Phi|^2(x) $ is less than some constant or $ L^n $ norm of $ |\Phi| $ is less than some constant. This is a generalization of the results on complete minimal submanifolds.
- hyperbolic space,
- traceless second fundamental form,
- the first eigenvalue

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References(17)

References

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